Differentiation - 01

So, what is differentiation???

Differentiation is the proces of finding the derivative of a particular function. And what is a derivative?

It means the rate of change of a particular variable (dependent), with respect to another variable (independent). For eg. - the rate of change of displacement with respect to time is velocity. This can be denoted by dx/dt = v

.......in this equation,x is the distance, t is time, v is velocity & d/dt is known as the operator of differentiation.

Graphically, this represents the slope of the tangent of the line at any particular point on it.Below is a figure showing the same.

In this figure, at the red dot, if differentiation is done of the curve shown in black, we would obtain the equation of the slope of the curve at that point i.e. you would get the equation of the red line. This means that it describes the best linear approximation of the function near that input value.

For this reason, the derivative is often described as the "instantaneous rate of change".

Now, let us bring the second point (x2,y2) closer to (x1,y1). As we bring it closer and closer, we find that the slope of m at the point x1,y1 is equal to slight change in y divided by slight change in x. This is what is meant as differentiation.